Astronomy
&
Astrophysics
A&A 535, A12 (2011)
DOI: 10.1051/0004-6361/201117748
c ESO 2011
Shock-induced polarized hydrogen emission lines
in the Mira star o Ceti
N. Fabas1 , A. Lèbre1 , and D. Gillet2
1
2
LUPM, UMR 5299, Université Montpellier II/CNRS, France
e-mail: [email protected]
Observatoire de Haute-Provence, USR 2207, OAMP/CNRS, France
Received 21 July 2011 / Accepted 8 September 2011
ABSTRACT
Context. In the spectra of variable pulsating stars, especially Mira stars, the detection of intense hydrogen emission lines has been
explained by the presence of a radiative and hypersonic shock wave, periodically propagating throughout the stellar atmosphere.
Previous observation of the Mira star o Ceti around one of its brightest maximum light led to the detection of a strong level of linear
polarization associated to Balmer emissions, although the origin of this phenomenon is not fully explained yet.
Aims. With the help of spectropolarimetry, we propose to investigate the nature of shock waves propagating throughout the stellar
atmosphere and present, for o Ceti (the prototype of Mira stars), a full observational study of hydrogen emission lines formed in the
radiative region of such a shock.
Methods. Using the instrument NARVAL mounted on the Télescope Bernard Lyot (TBL) in Pic du Midi Observatory (France), we
performed a spectropolarimetric monitoring of o Ceti during three consecutive pulsation cycles. For this survey, the four Stokes
parameters (I for intensity, Q and U for linear polarization, and V for circular polarization) were systematically collected, with a
particular emphasis on the maxima of luminosity, i.e. when a radiative shock wave is supposed to emerge from the photosphere and
starts to propagate outward.
Results. On hydrogen Balmer lines, over a large part of the luminosity cycle, we report clear detection of polarimetric structures in Q
and U Stokes spectra (and also in V Stokes spectra but to a lesser extent). We report a temporal evolution of these spectropolarimetric
signatures, which appear strongly correlated to the presence of an intense shock wave responsible for the hydrogen emission lines.
We establish that the hydrogen lines are polarized by a physical process inherent to the mechanism responsible for the emission line
formation: the shock wave itself. Two mechanisms are thus considered: a global one that implies a polarization induced by some giant
convective cells located around the photosphere and a local one that implies a charge separation due to the passage of the shock wave,
inducing an electrical current. Combined with the existing turbulence, this may generate a magnetic field, hence polarization.
Key words. polarization – shock waves – line: formation – stars: atmospheres – stars: individual: o Ceti
1. Introduction
Mira stars are cool, evolved and variable stars. Its prototype is
o Ceti (as known as Mira), which is the object of this paper.
On a Hertzsprung-Russell diagram, Mira stars are associated to
the asymptotic giant branch (AGB). Indeed, their main-sequence
mass is less than ten solar masses, which allows them to reach
the AGB phase where hydrogen and helium burn in shells around
a core of carbon and oxygen. They undergo radial pulsations
whose origin is explained by several theories, the main one being a supposed κ-mechanism (Cox 1980), which leads to a cycle of visual light variations. Some self-excited pulsation models of Miras already exist (Ireland et al. 2008 and references
therein), but the mechanism responsible for the pulsation has
not been identified yet. The free input parameters are mass, luminosity, composition, microturbulent velocity, mixing length
and turbulent viscosity. Using a non-local time-dependent theory of turbulent convection, Xiong et al. (1998) showed that the
pulsational stability of Mira stars is governed by the dynamic
and thermodynamic couplings between convection and oscillations. Xiong & Deng (2007) found that turbulent pressure plays
a key role for the excitation mechanism. The typical period of
Based on spectropolarimetric observations obtained at the
Télescope Bernard Lyot (TBL at Observatoire du Pic du Midi, CNRS
and Université de Toulouse, France).
Mira stars’ pulsation extends from 150 to 1000 days, the period
during which the luminosity might change up to 2.5 visual magnitudes on average. It has to be noted that those stars are not
as regular as, e.g., RR Lyrae. Therefore the duration of a cycle
might vary with time.
The low surface temperature of those stars (∼3000 K) allows for the existence of molecules in the very extended atmosphere and in the circumstellar envelope, all of which are
typical features of late-type stars. For the Mira stars of the
M spectral type, the most present molecules are oxygen-rich
ones such as H2 O, TiO, or CO. These molecules are at the origin of the strong molecular absorption undergone by the blackbody spectrum at this temperature (Tsuji 1966; Tsuji 1971a;
Tsuji 1971b; Jørgensen 1994; and Gillet 1988). This absorption
is such that the observed spectrum departs a lot from the corresponding black-body spectrum. Furthermore, and especially for
Mira stars, strong emission has been observed on Balmer hydrogen lines, lasting during up to 80% of the luminosity period.
In past decades, several studies – based on high resolution
spectroscopic surveys – have focussed on the atmospheric dynamics of several classes of radially pulsating stars located along
the instability strip: from the population II cepheid W Vir stars
(Lèbre & Gillet 1992) up to the RV Tauri stars (Lèbre & Gillet
1991), the Mira stars (Alvarez et al. 2000, 2001), and the postAGB stars (Lèbre et al. 1996). It is now admitted that radially
Article published by EDP Sciences
A12, page 1 of 10
A&A 535, A12 (2011)
pulsating variable stars host radiative hypersonic shock waves
that propagate periodically (the period is hardly changing with
time) throughout the stellar atmosphere.
Two main properties of these shock waves have been investigated: (i) their radiative nature, generating emission lines formation process (hydrogen and helium lines, see Gillet et al. 1983;
fluorescent lines, see Willson 1976); and (ii) their hypersonic nature, from an estimation of their propagation velocity that always
appears higher than the sound velocity in such media (several
Mach numbers; see Gillet et al. 1998a, and references therein).
Their structure has also been investigated (see Fadeyev & Gillet
2004, and references therein), and three main areas constitute
those radiative shock waves. First of all, the radiative precursor
is the area before the shock where the matter has not yet been affected by the raising wave compression. Then comes the shock
front. In a length of only several mean free paths, a part of the
kinetic energy is brutally converted into thermal energy, which
creates strong temperature, pressure, and velocity gradients. At
the end, the radiative wake is located behind the front. There,
the partially ionized matter gets cooler and recombines. The intense emission lines observed in the spectra of radially pulsating
variable stars, such as Mira stars, are supposed to be formed just
behind the shock front (Gillet et al. 1983) in the so-called radiative wake of the shock.
These studies have also related the shock wave propagation
phenomenon in the atmospheres of variable stars to their pulsating process (which is the κ-mechanism for the large majority of
these objects) or to more chaotic behaviour (for RV Tauri and
Post-AGB stars, for example, Barthès et al. 2000). These works
have mainly established that shock waves, which are connected
to the pulsation mechanism, are emerging from the photosphere
once a cycle, around the maximum light, and are thus propagating periodically throughout stellar atmosphere, ruling its dynamics, and extending it to several tens of solar radii, progressively
generating, and shaping a circumstellar envelope around coolest
objects.
However, to date, the magnetic nature of these shock waves
has never been identified or deeply investigated, while since the
60s, theoretical studies of shock waves propagating throughout
gaseous media of atmospheric density predict there is an electric field (therefore a magnetic field) in the post-shock region
(see for instance: Jaffrin 1965; Lu & Huang 1974; Vidal et al.
1995). Moreover, because of the complexity of the problem, this
magnetic nature has never been introduced in shock structure or
shock propagation modelling. Also, its impact on astrophysical
processes, such as radial pulsation or chaotic behaviour, atmospheric dynamics, circumstellar envelope formation and structure, mass loss, and grain formation, has never been investigated
so far.
In the late 70s, narrow- and broad-band polarimetric observations were performed on cool evolved stars hosting circumstellar envelopes in order to investigate the grain distribution in
stellar environments. With this technique, in a study devoted to
the one-year-period variable Mira star, o Ceti, McLean & Coyne
(1978) were the first to detect a strongly linear polarized flux in
the emission lines of the Balmer series. Based on just a single
observation limited to the maximum luminosity, the polarization
in the hydrogen lines (from Hβ to H10) was noticed as two to
three times greater than the polarization measured in the nearby
continumm. To date, this observational fact is still an enigma. It
has also never been reproduced on any Mira star or on another
pulsating star, or even characterized during a complete pulsating
cycle. Moreover, recent modelling of the linear polarization of
optical emission in red giants (Fadeyev 2007) which investigates
A12, page 2 of 10
light scattering by dust grains in axisymmetric circumstellar envelopes, has even failed to explain such a high polarization rate.
Recently, a new generation of instruments has been made
available: spectropolarimeters. Over a large spectral domain,
they combine high-resolution spectroscopy and information in
polarized light, and they open new observational windows for astrophysical objects, mainly focussing on stellar magnetism. With
these instruments, it is now possible to collect spectral information through the four Stokes parameters (V, characterizing circular polarization; Q and U, characterizing linear polarization;
I, the classical spectrum).
Harrington & Kuhn (2009) present spectropolarimetric observations of Post-AGB stars and RV Tauri stars and report, for
most of these objects, the detection of linear (Stokes Q and
Stokes U) polarimetric signatures associated to the Hα lines.
In most objects, the morphology of the polarization feature
presents short timescale variability that suggests the presence
of absorbing polarizing gas close to the star, according to the
optical pumping model described by Kuhn et al. (2007). Stong
mass loss, progressively shaping circumstellar environments, is
very common in these stars, and the shock waves propagating
throughout their atmosphere and inducing ballistic motions has
also been reported (Lèbre & Gillet 1991; Lèbre et al. 1996).
It is thus now possible for the very first time to estimate
the linear and the circular polarization evolution in the emission lines of Mira stars. As we are convinced that the hydrogen
emission lines are polarized by a physical process inherent to the
mechanism responsible for the emission line formation (i.e., the
shock wave itself), a spectropolarimetric survey performed during the cycle of pulsation of Mira stars appears to be a good tool
for tackling this problem. In Sect. 2, we present our observations
of o Ceti as collected with the NARVAL spectropolarimeter. In
Sect. 3, we interpret the obtained results in term of shock wave
characterization. In Sect. 4, we draw a conclusion from this spectropolarimetric analysis.
2. Spectropolarimetric monitoring of o Ceti
2.1. Observations
We performed a spectropolarimetric monitoring along three consecutive pulsation cycles of the bright star o Ceti, the prototype
of oxygen-rich Mira stars. During one of its typical cycles of
luminosity (with a mean period of pulsation of 331 days), the
visual magnitude of o Ceti presents an average variation amplitude of eight and its spectral type varies from M5e to M9e
(see General Catalogue of Variable Stars and AAVSO website). Omicron Ceti has the peculiarity of hosting a companion (VZ Ceti) at ∼70 AU believed to be either a white dwarf
(Karovska et al. 1997; Karovska 2006) or a main sequence star
(Ireland et al. 2007). We consider it to be too far to have any
influence on the dynamics of the lower atmosphere of o Ceti.
The spectropolarimetric data come from the NARVAL instrument mounted on the Télescope Bernard Lyot (TBL) in the
Pic du Midi Observatory, France. The polarimeter is made with
optical elements (a series of half-wave, quarter-wave, and halfwave rhombs) that analyse the circular or linear polarization
state of a signal (Donati et al. 1997). The processed signal is
thus transmitted in an achromatic way via an optical fiber to the
cross-dispersion spectrometer doing the spectral analysis over a
very wide part of the spectrum (from 375 to 1050 nm) and over
40 orders. The polarimetric mode gives us a spectral resolution
of 68 000, and the velocity bin is then Δv ∼ 4.4 km s−1 .
N. Fabas et al.: Polarized hydrogen emission lines in o Ceti
Table 1. Journal of our spectropolarimetric observations of o Ceti with
Narval.
Fig. 1. Light curve of o Ceti (from AAVSO website) for the three cycles
of the survey. Each blue cross is one spectropolarimetric observation of
our survey.
The monitoring was from the 5 September 2007 to the
10 February 2010, covering three luminosity cycles of o Ceti.
To provide phases during these cycles, we used as heliocentric
julian date of reference at luminosity maximum (see AAVSO
website): HJDr = 2 454 150. The phase ϕ is then the ratio between HJDo − HJDr and the period (see Fig. 1 and Table 1).
2.2. Data treatment
The Libre-ESpRIT software (Donati et al. 1997) processing data
collected by NARVAL allows us to obtain polarimetric spectra.
The outputs of Libre-ESpRIT process are the four Stokes parameters divided by a pseudo-continuum:
I
– the intensity I ,
c
– the linear polarization
Q I0◦ − I90◦
=
Ic
Ic
U
I45◦ − I135◦
u =
=
,
Ic
Ic
q =
– the circular polarization
v=
V
I − I
=
Ic
Ic
(I : left-handed circular polarization, I : right-handed circular polarization).
From this information, we can deduce the rate and the angle of
linear polarization (see Bagnulo et al. 2009):
Q2 + U 2
U
plin
1
=
, θ = arctan
+C
Ic
Ic
2
Q
⎧
⎪
if Q and U > 0
C = 0◦
⎪
⎪
⎨
◦
if Q < 0
C
=
90
with ⎪
⎪
⎪
⎩ C = 180◦ if Q > 0 and U < 0.
The Libre-ESpRIT software automatically corrects a pseudocontinuum. It deduces a black-body continuum from the observed spectrum by which this spectrum is divided. This feature is quite a problem in our case because the spectra of cool
Mira stars are very affected by molecular absorption. Moreover,
Date
Phase
HJD
Exp. Stokes S /N a
(dd/mm/yyyy)
ϕ
(2 454 000+) time(s) param.
348.63529
1200
V
29/277
05/09/2007
0.58 348.65190
1200
Q
27/268
348.66850
1200
U
31/278
486.23594
120
Q
139/704
486.24927
120
U
134/706
486.25322
120
V
136/707
486.25732
120
Q
140/695
120
U
135/690
20/01/2008
1.00 486.26126
486.26520
120
V
125/640
486.27026
120
Q
142/723
486.27420
120
U
144/741
486.27817
120
V
134/681
507.25895
120
Q
181/929
10/02/2008
1.06 507.26289
120
U
175/931
507.26683
120
V
179/945
708.64006
1200
Q
22/302
29/08/2008
1.68 708.65652
1200
U
21/288
708.67299
1200
V
23/304
889.26324
120
Q
132/525
889.26716
120
U
122/506
889.27113
120
V
130/538
26/02/2009
2.23
889.27522
120
Q
133/514
889.27914
120
U
132/536
889.28309
120
V
100/415
24/07/2009
2.67 1037.63799
520
V
16/223
1098.49282
400
Q
15/212
23/09/2009
2.86 1098.50000
400
U
17/208
1098.50718
400
V
15/204
1108.44130
400
Q
28/331
03/10/2009
2.89 1108.44847
400
U
32/367
1108.45565
400
V
28/324
1130.50330
120
Q
109/575
25/10/2009
2.95 1130.50725
120
U
102/537
1130.51119
120
V
104/549
1160.32884
40
Q
57/327
24/11/2009
3.05 1160.33185
40
U
55
1160.33487
40
V
56/337
1176.35850
160
Q
135/695
10/12/2009
3.09 1176.36289
160
U
138/700
1176.36728
160
V
150/759
1186.26722
28
Q
61/350
20/12/2009
3.12 1186.27008
28
U
59/334
1186.27294
28
V
58/329
1203.35451
40
Q
41/310
06/01/2010
3.18 1203.35750
40
U
42/311
1203.36053
40
V
40/285
1215.30800
160
Q
116/735
18/01/2010
3.21 1215.31242
160
U
112/715
1215.31681
160
V
111/707
1238.27729
520
Q
66/495
10/02/2010
3.28 1238.28585
520
U
47/355
1238.29443
520
V
44/344
Notes. (a) The signal-to-noise ratio refers to the Stokes I parameter at
412 nm and 647 nm, respectively.
because of the pulsational mechanism, the effective temperature presents a strong variation in the luminosity cycle (as well
as the associated black-body radiation). However, the emission
features that we are analysing are often very intense (hence
much higher than the local continuum). In the blue part of the
spectrum, where most of the Balmer lines are, the molecular
A12, page 3 of 10
A&A 535, A12 (2011)
Fig. 2. Hydrogen Balmer line signatures of the third cycle of our monitoring of o Ceti in Hγ (full Stokes). Successive signatures are artificially
offset vertically for legibility. Phases are indicated on the right (see Table 1).
absorption is less, and the continuum change is assumed to
be negligible with respect to the extent of the Balmer lines.
However, we chose to use non-normalized data and a pseudocontinuum defined by ourselves. A polynomial fit of second order is calculated on the pseudo-continuum around the line and
between –200 km s−1 and 200 km s−1 . Then, we normalize the
data with this fit.
The polarization rates given in our study might reach unexpectedly high values (several dozen percent). One could say that
no polarizing mechanism may be able to polarize light as efficiently, but it has to be borne in mind that this rate is given as a
percentage of the pseudo-continuum intensity, i.e. not as a percentage of the total intensity (where the total intensity meant as
the sum of the pseudo-continuum intensity and the intensity in
the strong emission line), which is what is usually given in theoretical works. This is the first time (except in McLean & Coyne
1978) that a polarimetric study is presented by analysing such
intense emission.
Compared to the work presented in McLean & Coyne (1978)
and also devoted to o Ceti, our study is based on much more
polarimetric observations collected at various phases. Since we
present the evolution of the intensity and polarization with respect to the pseudo-continuum, those observations made through
time are thus believed to be comparable.
A threshold of detectability can be set at 10−4 (Morin, priv.
comm.). Also, a limiting constraint is the crosstalk between different Stokes parameters owing to instrumental problems. It has
been measured to be no more than 1% (see NARVAL website).
Knowing that, we chose 3% as a reasonable limit for polarimetric signatures to be considered as detections.
A12, page 4 of 10
2.3. Spectropolarimetric data around Balmer lines
Figures 2 to 5 present the polarimetric data (in Hγ and Hδ) of
o Ceti for the best-sampled cycle of our survey (the third cycle).
Our analysis is focussed on those two Balmer lines, which are
the least affected by TiO absorptions. In Figs. 2 and 4, the spectra
of the four Stokes parameters I, Q, U, and V are displayed. The
abscissa is taken as velocity in the stellar rest frame (o Ceti’s
radial velocity: V∗ = 63.8 km s−1 , Wilson 1963), and the spectra
were convolved with a Gaussian curve of FWHM = 4 km s−1
to slightly reduce noise and improve the quality of the data. In
Figs. 3 and 5, we also show u(q), which can be read as a polar
graph with the linear polarization rate plin as the radial value and
2θ as the angle.
2.3.1. Hγ
From Figs. 2 and 3, devoted to the data around the Balmer line
Hγ, the emission appears at ϕ = 2.95, hence slightly before
the maximum light. During the full duration of our monitoring,
the Hγ emission line increases steadily and displays some well
known features (Joy 1947; Gillet 1988, Castelaz et al. 2000).
Its time evolution can be related to the presence of a shock
wave emerging from the photosphere just before the maximum
light and starting to propagate outward. While it first encounters a medium of decreasing density in the lower part of the
atmosphere, the shock thus accelerates, before being progressively dimmed in the highest atmospheric layers because of the
energy lost through radiation. While it is doing so, the column densities of absorbants (located above the shock’s front)
decreases slightly. Those molecular absorptions are shaping the
emission line to leave three peaks that can be easily identified
N. Fabas et al.: Polarized hydrogen emission lines in o Ceti
Fig. 3. Hydrogen Balmer line signatures of the third cycle of our monitoring of o Ceti in Hγ (u(q)). Phases are indicated in the topleft of each
frame (see Table 1).
from ϕ = 2.95 to ϕ = 3.28. The most redshifted peak is constant
(in intensity) through time, in contrast to the two others. The
central peak evolves quite rapidly around the maximum light
and then remains constant in intensity until the end of our observations. The bluest peak develops progressively and reaches
ϕ = 3.28 at an intensity that is five times higher than the one displayed at ϕ = 2.95. On its blue edge,a shoulder is also present
and disappears around ϕ = 3.21. Each of these peaks is globally redshifted with time with hardly any change in the width.
Throughout our observations, the total width of this emission is
conserved through time at 90 km s−1 .
When the intensity starts to develop, the Stokes parameters
q and u both present strong signatures. Stokes q has a positive
value at phase ϕ = 2.89, 3.05. It becomes zero and then negative through time, reaching its full amplitude at ϕ = 3.18. All
the signatures in q spread over a velocity width equivalent to the
one of the intensity emission and the features often present a line
reversal at their centre. The Stokes u signature remains positive
throughout our monitoring. It reaches its maximum amplitude
at ϕ = 3.09 and remains quite constant in shape and in amplitude until ϕ = 3.12. After the maximum light, the signatures
on u present a strong change in shape, because strongly asymmetric and because clearly becoming narrower with time. In a
less definite way, Stokes v signatures are very weak and present
structure variations through time. While the radiation is clearly
getting linearly polarized during the shock’s propagation (from
ϕ = 3.05 to ϕ = 3.28), circular polarization is hardly detected
except at the very end of our monitoring.
The variation in the angle of linear polarization is quite obvious when looking at the u(q) plots (Fig. 3). At ϕ = 3.05, the first
phase of definite polarization, θ is around 20◦ . Afterwards the
direction of linear polarization rotates counterclockwise to reach
90◦ at ϕ = 3.28. After the maximum light, at phase ϕ = 3.05, the
linear polarization rate reaches 0.2 and stays around this value
except for the last observation at ϕ = 3.28, where it decreases to
0.1. However, noticeable is that, when the intensity emission is
still growing, the linear polarization rate decreases after its maximum. Whatever is polarizing light works less efficiently at this
specific moment.
2.3.2. Hδ
From Figs. 4 and 5, which displays the data around the Balmer
line Hδ, the intensity emission appears at ϕ = 2.95 similar to
what is observed for Hγ. It starts to develop rapidly after the
maximum light and continuously increases during the monitoring to reach a maximum amplitude from ϕ = 3.21 to ϕ = 3.28.
The maximum intensity of Hδ is much higher than for Hγ.
However, the temporal evolution of Hδ is different in the sense
that the intensity stays constant during the last two phases in contrast to what happens in Hγ. This is probably because Hδ photons are the most energetic. Since the shock is weaker at that
time, they should be the ones whose production should start to
be reduced first, and Hδ is also less mutilated by molecular absorption than the Hγ line. One strong absorption at zero velocity
severely affects the profile, while another one is barely noticeable on the blue wing around maximum light. The total width
of the emission is constant through time at 75 km s−1 , which is
smaller than the one in Hγ. For this line, the width and position
are barely modified through time.
As in Hγ, from the maximum light up to the end of our observations, signatures are clearly detected in Stokes u, q, but also
in v. Stokes q varies from being slightly positive before the maximum in luminosity, to nearly cancellation around ϕ = 2.95,
and then to negative values from ϕ = 3.05. Then it starts to develop and is at a maximum from ϕ = 3.18 until the end of our
survey. The q signatures become narrow progressively and even
present a small peak appearing at some phases on the bluest side.
Stokes u is undergoing dramatic changes. After being roughly
zero until ϕ = 3.05, it develops positively, getting as high as
0.3 at ϕ = 3.09. Then this positive peak recedes while a negative structure develops, and finally at ϕ = 3.28 u signature is
fully negative. Over a large portion of our monitoring, Stokes u
presents an “inverse P Cygni” structure, positive in the bluest
part. This part of the spectrum corresponds to the one in intensity where the emission is the brightest. At last, the Stokes v
parameter presents a very strong feature at phases ϕ = 3.12 and
3.18, which almost disappears suddenly at following phases.
The evolution of the linear polarization angle in the u(q)
plane is slightly similar to Hγ. For phases such as ϕ = 3.05 and
ϕ = 3.12, the angle cannot be determined easily. However, we
can definitely state that θ evolves from 45◦ at ϕ=3.09 to eventually settle down at approximately 100◦ at the last three observations (from ϕ = 3.18 to ϕ = 3.28). This is interesting when
compared to Hγ where θ does not settle down. The linear polarization rate is at first as low as 0.1 at ϕ = 2.89 and clearer at
ϕ = 3.05. It increases up to 0.5 at ϕ = 3.18 and 3.21 and then
decreases to 0.4. This reduction in polarization seems to occur
during the shock’s weakening.
During those phases, Stokes v shows up before θ settles down
to some value. We know that the decrease in the intensity emission is due to the deceleration of the shock wave. In Gillet et al.
(1983), it is shown that the emission line in Hα increases until ϕ = 0.40 (the reference maximum is of course different than
ours) and then goes down. The change in the polarization behaviour described above thus certainly happens at the moment
of this deceleration.
2.3.3. Comparisons to previous cycles
In Figs. 6 to 9, we present some phases from the previous, but
less well-sampled cycles. When we compare the same phases
of different cycles, the intensities of Hγ and Hδ are roughly the
same, in amplitude, width, and shape. Therefore, we can safely
state that, for the considered phases of each cycle, the shock
wave will produce intensity emissions of similar strength.
In contrast, the u, q, and v Stokes parameters can strongly
vary from one cycle to another. Both symmetries and amplitudes
of q, u, and v signatures are found to be altered in many cases,
inducing a change in the linear polarization angle. Interestingly,
the linear polarization rate seems to be roughly conserved
A12, page 5 of 10
A&A 535, A12 (2011)
Fig. 4. Hydrogen Balmer line signatures of the third cycle of our monitoring of o Ceti in Hδ (full Stokes). See Fig. 2.
Fig. 5. Hydrogen Balmer line signatures of the third cycle of our monitoring of o Ceti in Hδ (u(q)). See Fig. 3.
through cycles. This allows us to stress the non-repeatability of
the polarizing processes in action. Indeed, we know already that
each cycle of the luminosity variation of o Ceti is not an exact
copy of the previous ones (see AAVSO website). Considering
that the shock wave is the consequence of the pulsation of the
star, as the luminosity variation is, then it is no wonder that the
polarizing mechanism linked to the shock wave does not behave
exactly the same as before. However, as we noticed for the third
cycle, the signatures’ widths do not change from one cycle to
another for a given Balmer line.
3. Interpretation
The appearance of an emission line indicates that the specific intensity of the back-lighting radiation (creating the continuum) is
less than the source function inside the shock. This is linked to
the intrinsic photon emission behind the shock’s front. Emission
A12, page 6 of 10
lines can thus help in determining the properties of this area.
Polarization also happens to be much stronger in the Balmer
lines than in the pseudo-continuum around them. Its temporal
evolution follows the evolution of intensity emission. It is therefore believed that the polarizing mechanism is also intrinsic to
the shock wave.
In Figs. 2 to 5, the evolution of intensity emission lines
Hδ and Hγ is quite similar, leading us to think that the shock
emerges from the photosphere just before the maximum light
and is getting stronger toward the end of the third cycle of our
survey. For both Balmer lines, the emission’s width and central
velocity (expressed in stellar rest frame) do not vary with time as
previously said. This would imply that the shock is not accelerating much. Moreover, if the width of the emission remains the
same, it probably means that the mechanism behind line broadening (micro-turbulence?) does not evolve with time. Willson
(1976) already addressed the characterization of shock wave
properties through emission lines width in Miras. In the case
of an optically thin medium and when broadening line mechanisms are not dominant, the width is an indicator of the flow
velocity where the radiation is emitted. When the gas is optically thick, we cannot give anything else but an upper limit for
this velocity. Moreover, the model of shock wave we chose also
influences this relationship. In Fadeyev & Gillet (2004), shock
waves models are used to create synthetic Balmer lines with different values of temperature, density and upstream velocity. The
models show a line doubling that is particularly obvious for Hα
and just slightly obvious for Hβ while Hγ stays single-peaked.
On the other hand, the widths of the emissions vary clearly with
density and upstream velocity.
A few shock-related mechanisms of polarization are probably operating. First of all, we consider a photospheric origin. The shock starts deep in the atmosphere, probably even
N. Fabas et al.: Polarized hydrogen emission lines in o Ceti
Fig. 6. Hydrogen Balmer line signatures of cycles 1 (thick lines) and 2 (thin lines) of our monitoring of o Ceti in Hγ (full Stokes). See Fig. 2.
Fig. 7. Hydrogen Balmer line signatures of cycles 1 and 2 of our monitoring of o Ceti in Hγ (u(q)). See Fig. 3.
below the photosphere. The photospheric origin has already been
thought of for polarization (e.g., Harrington 1969) but never
with a shock wave crossing it. Besides this, it is known that
stars with low gravity, such as Mira stars, have large convective cells. In Chiavassa et al. (2009), 3D models of red supergiants (RSG) photosphere were compared to interferometric observations in the near infrared. They conclude that RSG have
large-scale granulations undergoing a temporal variation. This
kind of work has also been done for AGB stars (Freytag &
Höfner 2008). Thus, this convective structure could transfer its
asymmetry to the propagating shock wave. Since the convective
cells are very large, the asymmetry is not cancelled by integration on the stellar disk, and the shock wave may be aspherical.
However, other sources of asymmetry (e.g., high-layer asymmetries) might be of interest in this respect (Ragland et al. 2008;
Pluzhnik et al. 2009; Chiavassa et al. 2010; Wittkowski et al.
2011). Fadeyev & Gillet (2001) have shown that the compression rate η behind a radiative shock can be written as
Ein − Ein1
FR − FR1
η=4+3
+3
(1)
Et
ρUEt
where Et is the specific energy in the translational degrees of
freedom, Ein the specific energy of excitation and ionization of
atoms, and FR the radiation flux. The subscript 1 refers to the
values in the state ahead of the shock. Thus in the presence of
radiation, the gas compression ratio can reach values of tens or
even hundreds, significantly greater than the compression ratio
without radiation (η = 4). With such high compression ratios,
the level of turbulence of the gas must be strongly amplified (see
Gillet et al. 1998b). When the shock wave passes through the
convection zone located around the photosphere, the convective
cells are thus sharply compressed and heated. From the point of
view of the shock, the convective structure is almost static. It is
likely that the convective field is disrupted and even destroyed
temporarily during the passage of the wave. Consequently, if the
heterogeneous distribution of convective cells causes polarization, then it should also be observed in the continuum and be
disturbed during the photospheric passage of shock. It seems
that the quality and temporal resolution of current observations
are not yet sufficient to confirm this point directly. New observations are needed.
More local phenomena can be called upon. When the Mach
number of shock wave increases, its structure quickly becomes
A12, page 7 of 10
A&A 535, A12 (2011)
Fig. 8. Hydrogen Balmer line signatures of cycles 1 (thick lines) and 2 (thin lines) of our monitoring of o Ceti in Hδ (full Stokes). See Fig. 2.
appearing later in the wake when the coupling with the radiation field can be fundamental. As discussed above, the solenoidal
state of the postshock gas strongly increases in the presence
of a radiative field (shock turbulence amplification, see Gillet
et al. 1998b). Consequently, because the shock propagates in a
non-uniform background plasma, a spontaneous generation of a
magnetic field is possible within the postshock plasma. From the
induction equation, the equation describing the development of
magnetic field is
∂B
= ∇ × (v × B) + η∇2 B + S .
∂t
Fig. 9. Hydrogen Balmer lines signatures of cycles 1 and 2 of our monitoring of o Ceti in Hδ (u(q)). See Fig. 3.
complex. In particular, the charge separation within the shock
front region may induce an electric field. The nature of the resultant interaction of the electric field with the gas flow has been investigated for a steady plane shock in fully and partially ionized
gases (see for instance Jaffrin 1965; Lu & Huang 1974; Vidal
et al. 1995). The electric field is parallel to the direction of shock
propagation. However, the presence of background density or
temperature gradients, for instance, causes an electrical current,
inducing a curl of magnetic field. Thus the shockfront becomes
a source of magnetic field, which pervades the unperturbed gas
and the wake of the shock. These studies were limited to the
region of viscous shock and did not consider the plasma state
A12, page 8 of 10
(2)
The three terms on the right-hand side describe the convection,
diffusion, and generation of a magnetic field. The self-generation
magnetic field could be due to several local mechanisms such
as ion-electron separation at the front or density and temperature gradients in the plasma. Thus the dominant physical process must be determined to write the source term S. The magnetic field is perpendicular to the shock propagation direction.
Consequently, a linear polarization could be expected, although
because of the complexity of the environment, a minor circular
polarization could also be observed.
The observation of the polarization of emission lines has
been a long-standing issue in solar flare physics (Haug 1981),
in tokamak (Fujimoto et al. 1996), gasdischarge (Walden et al.
1999), and laser produced plasmas (Yoneda et al. 1997). The
emission polarization can come from different causes induced
by the non-uniform background plasma. In particular, it has
been known that anisotropic collisions of electrons with atoms
produce polarized radiations in plasma in which the collisional
excitation and ionization processes dominate (impact polarization). Thus, the generation of polarized emission can occur during the recombination processes. In fact, there are few theoretical
N. Fabas et al.: Polarized hydrogen emission lines in o Ceti
4. Conclusion
Table 2. Time evolution of spectropolarimetric values in Hγ.
ϕa
2.89 2.95 3.05 3.09 3.12 3.18
I/Icb
Q/Ic
U/Ic
V/Ic
plin /Ic
θ
0
0.1
0
0
0.1
20◦
6
0
0
0
0
0◦
12
0.1
0.2
0
0.2
40◦
3.21
3.28
14
14
15
20
26
0.05 –0.1 –0.2 –0.15 –0.15
0.22 0.2 0.15 0.2
0
0
0
0
0
–0.1
0.22 0.2 0.25 0.22 0.15
45◦ 45◦ 80◦
80◦
90◦
Notes. (a) The phase along our survey of the third cycle. (b) From lines 2
to 6 are the maximum values of the indicated parameter.
Table 3. Time evolution of spectropolarimetric values in Hδ (see
Table 2).
ϕ
2.89 2.95 3.05 3.09 3.12 3.18 3.21 3.28
I/Ic
5
8
38
55
60
65
80
80
Q/Ic 0.15
0
–0.15 –0.15 –0.25 –0.5 –0.5 –0.4
U/Ic
0
0
0.1
0.25
0.2 –0.2 0.1 –0.2
V/Ic
0
0
–0.1 –0.05 –0.26 –0.5
0
–0.1
plin /Ic 0.15
0
0.15 0.28 0.25 0.5 0.5 0.4
θ
20◦
0◦
90◦
45◦
90◦ 100◦ 90◦ 100◦
and experimental works about this matter, and devoted to radiation fronts, laboratory ionizing shock tubes, or radiative shock
waves. In conclusion, we expect that the observed polarization
in emission lines of Mira could have been caused by the spatial
non-uniformity in the electron distribution function that was induced by the self-generated magnetic field created in the shockfront. It is not possible within the framework of this study to
definitively explore the physical origin of this polarization. Only
a quantitative approach may allow us to answer.
Tables 2 and 3 sum up the evolution of spectropolarimetric values for Hγ and Hδ for the 3rd cycle. Considering only
I profiles, it is difficult to state at which moment of the cycle
the shock reaches its maximum intensity because the emissions
do not undergo the same molecular absorption. Then, polarization could help determine this moment. If we consider the local mechanism, the turbulence would be at its maximum during
phases of most intense emissions, inducing the strongest polarization. It is interesting to see that linear polarization appears
during a longer time than the circular one. Stokes V is rarer and
mostly present when the linear polarization is quite strong, i.e.
at about phase 0.20. Indeed, this is consistent because when the
shock becomes strong, the turbulence amplification is at a maximum, and consequently the hydrodynamic motions are more and
more tridimensional. It is interesting to note that phases around
0.20 are quite common in atmospheric models for merging shock
fronts (Ireland et al. 2011).
To know which of those two phenomena are likely to occur,
it is necessary to make similar observations of other types of
radially pulsating stars that have no convection, but are known
to host an atmospheric shock wave. The detection of polarization in the Balmer emission lines of such stars would then give
credit to the local hypothesis. On the other hand, if the polarization arises from the convective asymmetry, it would also act
on the continuum of the photosphere’s radiation. At that point,
the global hypothesis would prevail. A temporal survey of the
polarization of this continuum would then be necessary to see
whether it varies like the Balmer lines polarization presented in
Tables 2 and 3.
We performed the first spectropolarimetric monitoring of the
Mira star o Ceti, focussing on Balmer emission lines. Thanks to
these observations, we were able to detect variable linear and circular polarization signatures associated to those hydrogen lines.
The temporal variation of those signatures is correlated with
the evolution of the intensity of the emission lines known to
be formed in the radiative wake of a shock wave propagating
throughout o Ceti’s atmosphere.
Several potential polarizing mechanisms are considered.
AGB stars host very large photospheric convective cells inducing overhall asymmetry. Since the shock is formed near the
photosphere, it is possible for it to interact with the convective structure and thus to gain the photospheric asymmetry.
Interferometric observations at minimum light, i.e. when the
shock is high in the atmosphere, would help to compare the
photosphere’s and the shock’s geometrical features. On the other
hand, our local hypothesis states that a charge separation induced
by the shock’s passage along with the turbulence believed to exist behind the front of the shock could produce a local magnetic
field through which the radiation from the wake becomes polarized.
Unlike o Ceti, other Mira stars can be rich in carbon or have
no companion. They would thus be interesting to observe to determine how general this evolution of the line polarization is.
This study might also be extended to other kinds of pulsating
variables stars, such as RV Tauri or RR Lyrae, known to host radiative shock waves propagating throughout the atmosphere. If
pulsating stars with no convection (such as β CMa stars) were
still showing such line polarization, it would favour the local hypothesis.
Acknowledgements. We acknowledge with thanks the variable star observations of the AAVSO International Database contributed by observers worldwide and used in this research. We also acknowledge financial support from the
“Programme National de Physique Stellaire” (PNPS) of CNRS/INSU, France.
We are grateful to V. Bommier, A. Chiavassa, J.-F. Donati, J. Morin, and F.
Paletou for useful comments.
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